how do I modify my topology to follow the path I want instead of what it already is?
For proper Retopology, it is recommended to keep the original mesh as a mere foundation of the shape. And then start with a new object from scratch and apply the desired topology.

Here is a relatively easy way how to do good retoplogy without using any third party addons:


Step 1: Create your Base Model. The Toplogy is unimportant at this point. It also can be a sculpted mesh with a high polycount. But make sure it does not slow your system too much down:


Step 2: Create a new object and open the edit mode.

Step 3: Enable "snap during transform" and set it to "Face" 


Step 4: Select a vertice point and hit G. Snap it to the desired position.


Step 5: Form the desired shape of your topology.


Step 5: Continue to form the topology and connect edges.


Step 6: Go on until you are done.

Also here is a list of some good topology references: https://de.pinterest.com/demons_fate/mesh-toplogy/